Not sure for that exact scenario as I don't have it in front of me, but unless it says that the player is eliminated as soon as their Planeswalker is destroyed, they can continue playing with their other creatures. Unfortunately, without a Planeswalker, they can no longer cast spells or summon new creatures to the battlefield. You can still trigger face down cards that are already in play though.

Page 7 of the rulebook under Planeswalker Rules covers that. As I said, I don't remember the specifics of that scenario, but unless it specifically says otherwise, those core rules would still apply.